Delft University of Technology Faculty of Electrical Engineering, Mathematics and Computer Science Delft Institute of Applied Mathematics History Matching via Ensemble Kalman Filter of Norne Field A thesis submitted to the Delft Institute of Applied Mathematics in partial fulfillment of the requirements for the degree MASTER OF SCIENCE in APPLIED MATHEMATICS by Slawomir Pawe l Szklarz Delft, The Netherlands November 17, 2010 Copyright c ⃝ 2010 by Slawomir Pawel Szklarz. All rights reserved.
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Delft University of Technology
Faculty of Electrical Engineering, Mathematics and Computer Science
Delft Institute of Applied Mathematics
History Matching via Ensemble Kalman Filter of Norne Field
A thesis submitted to the
Delft Institute of Applied Mathematics
in partial fulfillment of the requirements
for the degree
MASTER OF SCIENCE
in
APPLIED MATHEMATICS
by
S lawomir Pawe l Szklarz
Delft, The Netherlands
November 17, 2010
Copyright c⃝ 2010 by S lawomir Pawe l Szklarz. All rights reserved.
MSc THESIS APPLIED MATHEMATICS
”History Matching via Ensemble Kalman Filter of Norne Field”
S lawomir Pawe l Szklarz
Delft University of Technology
Daily supervisor Responsible professor
Dr. Remus G. Hanea Prof. dr. ir. Arnold W. Heemink
where x0 is an initial covariance matrix. The aim is to use the available observations y to find
the best estimate of the state of the system x.
Figure 2.1: Methodology of the Ensemble Kalman Filter
Analysis Scheme
The Ensemble Kalman Filter at each simulation time consists of two steps, i.e. the fore-
cast step (or forward) in which the state vector is simulated in time and the analysis step
8 CHAPTER 2. TECHNIQUES, THEORY AND FORMULAS
(or assimilation step) where it is corrected to honour the observations. Both steps can be
done simultaneously. In the analysis procedure the model variables are updated using Kalman
equation. Both, static and dynamic variables are adjusted to honour observed data. All the
previously simulated estimates are stored in the ensemble.
In the reservoir engineering the forecast step is achieved using a reservoir simulator where
the physical system is approximated by a numerical model with discretization of a system of
differential equations. It is worth to note that simulation has to be done for every ensemble
member separately, however they can be done simultaneously and parallel computing can be
applied.
Denoting k as a current time step, the equation of the forward step takes the form of
xfk = f(xak−1) + wk (2.5)
Function f represents here the work done by the reservoir simulator. Defining now the mean
of the forecast ensemble as
Exfk =1
n
n∑i=1
xfk(i) (2.6)
and the matrix of differences
Lfk = [xfk(1)− Exfk , ..., x
fk(n)− Exfk ] (2.7)
ensemble covariance matrix can be expressed in terms of
P fk =
1
n− 1LfkL
fk
T(2.8)
The ensemble mean defined in equation (2.6) is interpreted as the best estimate of the state
and the spread of the ensemble reflects the associated uncertainty.
the second step is called analysis step as the forecast is conditioned on observations
xak = xfk +K(yk + vk − Ckxfk) (2.9)
From the last equation we can observe that the modification of the state vector depends on the
disagreement between simulated values and observations. The larger the difference, the larger
the adjustment is performed. Note that observations yk were perturbed as mentioned before
by the Gaussian white noise vk. The errors of measurements are assumed to be independent
and thus the covariance matrix is diagonal. The matrix denoted as K in equation (2.9) is so
called kalman gain matrix defined by
Kk = P fk C
Tk (CkP
fk C
Tk +Rk)
−1 (2.10)
2.5. SENSITIVITY ANALYSIS AND SEQUENTIAL GAUSSIAN SIMULATION 9
where the term Rk stands for the measurement error covariance matrix defined as
Rk =1
N − 1HkH
Tk (2.11)
with Hk storing perturbations of measurements. The ensemble covariance matrix takes the
form of
P ak = (I −KkCk)P
fk (2.12)
Simple Kalman Filter example
To illustrate that the Kalman gain matrix plays here simply a role of assigning a weight to the
model and observations, consider scalar time invariant model (one dimensional)
xk+1 = xk + wk (2.13)
yk = xk + vk (2.14)
and Q, R as variances of noise processes w and v, respectively. For sufficiently large k the
kalman gain matrix becomes constant
K∞ =1 +
√1 + 4R
Q
1 + 2RQ +
√1 + 4R
Q
(2.15)
The mean is then updated as follows
xak = xak−1 +K∞(yk − xak−1)
= (1−K∞)xak−1 +K∞yk (2.16)
It can be noticed in the equation (2.17) the linear combination between the simulated values
in the previous time step and measurements at the current time k. If the observations noise R
is very large relatively to the model noise Q then K∞ is closed to zero and new measurements
have little impact on the state. The other way around, K is near 1 thus observations get the
higher weight.
2.5 Sensitivity analysis and Sequential Gaussian Simulation
Before the EnKF method is applied, there is a need to choose parameters to update in the
procedure. From a number of parameters it is important to select the properties of the reser-
voir (usually porosity and permeability) that have the greater impact on the flow simulation
results. It can be done by slightly perturbing each parameter, then running a simulation and
obtaining reservoir response. Each parameter change requires separate flow simulation thus
the process can be time consuming.
When the parameters specified to create initial ensemble has been chosen the ensemble can
be created. Very popular method to simulate multivariate Gaussian field is sequential Gaus-
sian simulation(SeGS). The values are simulated sequentially and they are conditioned on the
10 CHAPTER 2. TECHNIQUES, THEORY AND FORMULAS
original data and previously simulated values in the neighbourhood of considered point. SeGS
consists of the following steps:
(1) Transform the data to standard normal distribution if needed
(2) Compute and model the variogram
(3) Specify the path of visiting points zi on the grid
(4) Apply at every point the procedure:
(4.1) Obtain the mean and the variance of the property
at the point zi via kriging with the variogram
(4.2) Draw the value from normal distribution with the input
from kriging and add the value to the dataset
(4.3) Go to the (4.1) and repeat for the next point zi+1
until all the points are finished
(5) Transform back from the normal distribution if there is a need to.
Kriging provides the best unbiased estimator of the unknown parameter at the considered
point. Such estimator is a linear combination of the values within the specified neighbour-
hood of the kriging point. The degree of relationship between the points is described by
semi-variogram. More detailed theory on kriging is presented by Brown and Falade[2003] [12].
2.6 Localization and inflation
Spurious Correlations
The EnKF method requires sufficient size of an ensemble to capture the statistics. Nowadays,
it is common to use about 100 members however it can be still computationally demanding in
several applications. Hence, the focus is on developing techniques which allow to reduce the
number of ensemble members. The limit of the size of an ensemble leads to sampling errors
represented by spurious correlations. It can happen either when variables are far away or they
are supposed to be uncorrelated. As a result the error covariance matrix is underestimated [13].
Localization
Evensen [13] describes the localization method to reduce impact of spurious correlations. Con-
sidering the model grid it can be assumed that the observations only in neighbourhood have
impact on analysis of considered grid point thus only them can be involve in computation.
Moreover, the advantage of this method is the reduction of the simulation time in case when
the measurement data is large comparing to the ensemble size.
From the analysis equation (2.9) in EnKF, denote so called innovation as
dk = yk − vk − Ckxfk . (2.17)
2.6. LOCALIZATION AND INFLATION 11
Using (2.8) and (2.11) we can write (2.9) in the form of
xak = xfk + LkLTkCk(CkLkL
TkC
Tk +HkH
Tk )
−1dk (2.18)
Introducing
sk = CkLkT and zk = sksk
T + (N − 1)Rk. (2.19)
it is possible to rewrite the analysis scheme (2.9) as
xak = xfk + Lk(I + sTk z−1k dk). (2.20)
Denoting the expression between brackets as Uk, the EnKF analysis simplifies to
xak = xfkUk. (2.21)
The calculations above, presented with more details in Evensen [13], show that the analysis
step equation in EnKF is acombination of the forecast ensemble members i.e. spanned by the
ensemble members.
Denoting visited gridpoint as j and considering Uk introduced before the local analysis scheme
for jth grid block goes as follows
xak,j = xfk,jUk,j (2.22)
= xfk,jUk + xfk,j(Uk,j − Uk) (2.23)
The Uk is a global analysis term whereas in Uk,j only observations within specified distance
from jth grid point are used and thus it refers to local analysis solution.
Inflation
There is another method described in Evensen [13] to deal with the spurious correlations,
namely, covariance inflation method. Underestimation of the covariance matrix can be im-
proved by modifying each ensemble forecast accordingly to the formula
xfk(i) = c(xfk(i)−Exfk) + Exfk (2.24)
where the index i represents the ensemble of interest and c is a constant greater than 1. It
was shown that setting c too large causes overweighting the measurements [14]. Hence, there
is several approaches to optimize the inflation factor [15] [16] [17].
Chapter 3
Reservoir engineering basics and Norne Field
3.1 Reservoir Engineering
Reservoir structure can in many cases consist of sand and shale layers. Each layer has specified
parameters like porosity and permeability. The porosity of the reservoir sands is about 10–30
%. Typical permeabilities are in the range 0.1 – 10 D. Reservoir must be bounded by caprocks
or impermeable faults which enclose oil and gas inside the reservoir. Usually there are three
main phases in a reservoir i.e. water, oil and gas. Additionally, sometimes there are two more
phases included. The gas which becomes fluid at a surface called vapour fluid and fluid that
becomes gas i.e. dissolved gas. Differences between their physical properties, i.e. in density,
determines their vertical location in the reservoir. According to this gas-oil (GOC) and oil-
water (WOC) contacts can be observed. The state of oil, gas and water is described in a
reservoir simulation model by pressure and saturations. Some basic nomenclature associated
with reservoir engineering is explained in this section with the emphasis on terms used further
in this paper.
Porosity and permeability are two factors that influence fluid movement and storage in rocks.
Porosity measures how much void space is in the rock. It is a fraction of the volume of
void space to the total volume. Thus it is a number between 0 and 1 or a percentage.
Mathematically speaking
Φ =Vp
Vb(3.1)
where Φ is the porosity, Vp is the pore volume and Vb is the bulk volume. Another important
reservoir property related to pore volume is phase saturation. It is a ratio of the volume of the
phase to the volume of the rock. Since three phases are considered in the reservoir it can be
13
14 CHAPTER 3. RESERVOIR ENGINEERING BASICS AND NORNE FIELD
expressed for each of them
Sphase =Vphase
Vp(3.2)
where Sfluid is fluid saturation (of oil So, water Sw or gas Sg) and Vp is the pore volume. It
is natural that So + Sw + Sg = 1 and Vo + Vw + Vg = Vp. Permeability is a measure of the
ease with which fluids will flow through a porous material. Widely use in reservoir engineering
Darcy law is of the form
u =q
Ac= −k
µ
d(p+ ρgh)
dl(3.3)
where u is fluid velocity[ cms ], Ac is cross sectional area of the rock[cm2], q is the flow rate
[ cm3
s ], k is notation of permeability[D](Darcy), µ is viscosity of the fluid[cP ](centipoises), dpdl
is pressure gradient in the direction of flow [atmcm ], ρ is fluid density, g gravitational constant
and h is vertical depth. Since most of the reservoirs have permeability smaller than 1 [D], the
more convenient unit is milidarcy [mD]. Permeability in the equation (3.3) is called absolute
permeability (the rock is saturated with only one fluid). In the case of multiple number of
fluids in the rock we consider effective permeabilities of oil ko, water kw and gas kg. The ratio
of effective permeability to the absolute is called relative permeability
kro =kok, krw =
kwk, krg =
kgk. (3.4)
In Darcy equation not yet explained term viscosity appears. It is defined as a measure of
resistance of fluid which is being deformed by a stress. Its unit is called poise and it is kgm s in
SI system.
Another meaningful reservoir property is compressibility which measures how the pore volume
changes due to the changes in pressure while the temperature is constant. It is given by the
equation
c = −1
v
dv
dp(3.5)
where c states for compressibility, v is the volume and p is the pressure.
Capillary pressure is the difference between two immiscible fluids across the interface. It is
generally defined as
pc = pnw − pw (3.6)
where pnw is called non-wetting phase and pw wetting phase. In oil-water systems, water is
usually the wetting phase, whereas in gas-oil systems it is oil.
3.2. RESERVOIR DATA 15
Another important reservoir property that influences oil reserves is net-to-gross (N/G) ratio.
Reservoir usually consist of sand and shale while shale reserves should not be included in final
reserve calculation. N/G discounts proportion of shale to the total volume. Thus it is a number
between 0 and 1.
3.2 Reservoir data
The oil, gas and water are acquiring in the reservoir from the wells called producers. One
of the possibilities to force fluid to travel to particular producer is water or gas injection into
an injection wells. This technique requires at the beginning to study the reservoir in order
to make a plan of setting up a network of drilled wells. After the well is drilled it can be
completed. Completion is the process of preparing well to be enable to produce or inject.
During the production after reaching an economic limit, which means that expenses of the
production from the particular well bore are higher then the cost of produced oil, the well
is usually abandoned and buried. Wells are also employed to collect the observations used
further in a reservoir and research study e.g the history matching process. The data obtained
from the well bore is called either the log data or the core data depending on whether it is
received by a sensor located in the well or read out at a surface laboratory from a sample core
removed from the reservoir. Usually the production data, the injection data and the bottom
hole pressure are measured or calculated. The rates data may consist of oil, gas and water
rates observed at the well locations in the reservoir for the defined periods of time for instance
days. The bottom hole pressure (BHP) is the pressure at the bottom of the well bore. It can
be either measured using special equipment or estimated using available techniques, e.g.
BHP = ρgh+ THP (3.7)
where ρ is fluid density [kg/m3], g is gravitational constant [m/s2], h stands for the vertical
depth of the well [m] and THP (tubing-head pressure) is the pressure measured near the
surface in the well [18]. The bottom hole pressure is an important tool to maintenance the
reservoir. If its value is less than formation pressure (pressure of fluid within the pores) than the
fluid can flow into the wellbore. There are costs related with the placement of each well thus
there is a constraint on number of wells in a reservoir. For the engineers it results in working
with locally available data and thus the uncertainty of the rock properties in the reservoir rises
with the distance from the well.
Another type of the data possible to obtain is seismic data. The signal from a source mounted
at a surface of the reservoir is partially reflecting from the layer surface and recorded again at
the top of the reservoir by receivers. Thus the method is very helpful in describing the layers
lithology and regional properties of the reservoir. In 3D seismic a large number of 2D parallel
lines of signals are shoot into reservoir and after repeating this experiment, called survey, the
4D seismic are acquired as the fourth dimension is time. Comparing it to the log data which
16 CHAPTER 3. RESERVOIR ENGINEERING BASICS AND NORNE FIELD
can be collected very often in time (even every few seconds) the surveys are usually done in a
distance of a years.
3.3 Norne Reservoir
Norne Field is located 200 km from Norwegian coast line in the geological blocks 6608/10 and
6508/1 in the Norwegian Sea. Structure size is approximately 3x9 km and sea depth in the area
is 380 m whereas reservoir depth is 2500 - 2700 m. It was discovered in December 1991. The
production of oil and gas started November 1997 and 2001 respectively. Reservoir is operated
by Statoil Hydro Petroleum AS (63.95%) and partners: Petoro (24.55%) and Eni Norge AS
(11.5%). Data is provided through Integrated Operations in the Petroleum Industry (IO). The
field parameters have quite good quality. Porosity is in the range of 25-30%, net-to-gross ratio
0.7 - 1 and permeability varies from 20 to 25000 mD. Reservoir thickness changes from 120
m to 260 m from south to north.
Figure 3.1: Location of Norne Field,Statoil(2001)
Figure 3.2: Structure of Norne,Statoil(2006)
The Norne Field consists of Main Structure (segments C, D and E) and G segment. About
98% of oil is located in Main Structure. Particular segments can be seen in the Figure 3.2.
Since reservoir consist of horizontal, geological formations, in the Figure 3.3 one can find
zonation table. Two the most important formations are Ile and Tofte where is 36% and 44%
of reservoir oil, respectively. Gas is accumulated in the Garn formation at the top of the
reservoir. Neighbouring picture demonstrates cross section through the reservoir and major
faults. Since gas is more lighter then oil and water, in the hydrostatic equilibrium one can find
gas above oil and oil above water. This induces placement of gas-oil and oil-water contact.
Oil is enclosed in the reservoir thanks to faults and the cap rock (Melke Formation) which
seals the reservoir. Not Formation prevents communication between Garn and Ile formation
thus the decision has been made of switching the gas injection at the top of the field to the
3.3. NORNE RESERVOIR 17
Figure 3.3: Zonation of Norne Field, Fawke(2008)
Figure 3.4: Development technique for Norne, Statoil(2004)
water injection at the bottom of the reservoir, see Figure 3.4. According to the Norwegian
Petroleum Directorate (2010) recoverable reserves in Norne Field are: 94.90 mill Sm3 of oil
and 11.00 bill Sm3 of gas. There is still 14.00 mill Sm3 of oil and 5.20 bill Sm3 of gas as a
remaining reserves. The details can be found in Statoil PL128-Norne Reservoir Management
Plan [2001]. [19] [20] [21]. There are two types of barriers that can prevent flow going through
the reservoir. Faults discovered mainly by studying seismic data can either facilitate or block
18 CHAPTER 3. RESERVOIR ENGINEERING BASICS AND NORNE FIELD
Figure 3.5: Cross section through Norne, Statoil(2006)
the flow of fluid in a reservoir. Except faults there are horizontal Stratigraphic Barriers in Norne
mainly at a contact places between neighbouring formations: Garn3/Garn2, Not Formation,
Ile3/Ile2, Ile2/Ile1, Ile1/Tofte4, Tofte2/Tofte1 and Tilje3/Tilje2.
Chapter 4
ECLIPSE model import into Jewel Suite
4.1 Norne Model in ECLIPSE
ECLIPSE from Schlumberger is one of the leading reservoir simulators in oil industry. It is a
batch program. As an input user creates text file with a set of keywords that must be located
in particular section. Such data file gives complete description of a reservoir. The following
section describes shortly the model built in ECLIPSE simulator. Detailed description of the
deck and the particular keywords can be found in Appendix A.
Figure 4.1: Norne model grid and E-segment
The Norne Field model starts at 06 November 1997. The dimensions are 46 × 112 × 22, the
unit system is metric and five phases gas, oil, water, dissolved oil and vapour gas are activated
in the simulation. The grid consists of 113344 cells, where 44927 are active cells and the
grid units are meters. Reservoir properties are assigned to every cell then they are modified
according to specific segments, wells and layers. Net-to-gross, porosity and permeability ap-
pears to have a layer-dependent structure. The defined permeability in X direction is copied
to Y direction and Z direction. However, permeability Z is reduced using multipliers according
to particular layer. This means that permeability in X and Y direction are the same while
permeability Z differs. Specified transmissibilities are modified further in the edit section to
honour the changes in a reservoir structure made by drilling through the faults and the layers.
Areas near the wells have set increased transmissibility multipliers. For Norne the values varies
from 0.00075 to 20. Transmissibility multipliers only for two faults are bigger than 1 what
means that appearing of this faults increased easy with which flow goes through that fault.
19
20 CHAPTER 4. ECLIPSE MODEL IMPORT INTO JEWEL SUITE
The reservoir can be subdivided into regions if there is a need to set different local prop-
erties for the field. There are 4 flux regions for each geological layer: Garn, Ile, Tofte, Tilje-top
and Tilje-bottom. Thus there is 20 regions in total in Norne Field. There are transmissibil-
ity multipliers specified between each pair of neighbouring regions. Since the interest is in
Figure 4.2: Localization of wells in E-segment
E-segment, only the wells located in considered area are typed. There is five wells in the E
segment, two injectors and three producers. Wells localization can be seen in the Figure 4.2.
If there is a need to separate a part of the model, i.e. when only a sector of the field is
history matched, the time required for the simulation can be reduced by use of flux or coarse
option. In the first case the full grid model is simulated and the flow rates between specified
Figure 4.3: Norne Field original grid model Figure 4.4: Norne Field coarsened grid model
region and the rest of the field are saved to the flux file. Then it is possible to perform reduced
run when only the region of interest is active and the boundary flow rates are read from the
previously created file. As an alternative, the coarse option allows to merge a number of cells
in a specified box which is treated further by a simulator as a cell. The properties previously
assigned to the cells in the box are up scaled. Original and a coarsened grid for Norne Field is
presented in the Figure 4.3 and Figure 4.4, respectively.
More informations on Norne can be found in Norne Comparative Case Study, SPE127538
[22] and also on the Norne website[23] where the dataset and publications on the topic are
available.
4.2 Norne Model transfer from ECLIPSE to Jewel Suite
JOA Oil & Gas Jewel Suite is a full workflow-integration framework which allows to build
and design from the seismic interpretation, the grid representation to static and dynamic
4.2. NORNE MODEL TRANSFER FROM ECLIPSE TO JEWEL SUITE 21
programming and the wells structure. It has a user friendly interface guiding step by step
through preparation of the model and its workflow. Reservoir simulators such SENSOR,
ECLIPSE and IMEX can be plugged in and the results of the simulation can be imported and
viewed or prepared for another simulation.
Figure 4.5: Cooperation between ECLIPSE and Jewelsuite
It can be seen in Figure 4.5 that integration of Jewel Suite with ECLIPSE requires as a first
step import of the ECLIPSE deck to Jewel Suite. Unfortunately not all of the keywords used
in Norne Field model were supported in Jewel Suite 2009. To avoid errors appearance in Jewel
Suite and to import data correctly these keywords were commented out in the deck. Fortu-
nately, in Jewel Suite there is an editor of the deck pushed out further to run by ECLIPSE. Thus
allows user to make so called journal edits to deck before it is going to be run in ECLIPSE.
Some of the commands were added again as journal edits but some of them were skipped due
to the reimplementation issues. The list of trouble keywords and their meaning is presented
in Table 4.1.
Keywords ZIPPY2 and PIMULTAB were added to the output deck in Jewel Suite thus they are
include in the simulation. ZIPPY2 controls time step length to prevent convergence errors and
PIMULTAB sets productivity index scaling factor for the well for particular maximum water
cut value. There is a few more keywords in the SCHEDULE section that were omitted and
they can be seen in the Table 4.2. Generally they control behaviour of the well network i.e.
pressure and field economic data. Some of them appears more than once for multiply number
of time steps i.e. GECON.
22 CHAPTER 4. ECLIPSE MODEL IMPORT INTO JEWEL SUITE
keyword section purpose
TRACERS RUNSPEC specifies dimensions for tracers required to allo-cate memory
TRACER PROPS used to define tracers and their propertiesWTRACER SCHEDULE defines value of concentration in the related phase
for an injectorDRSDT SCHEDULE with this keyword user sets maximum rate of gas-
oil ratio increase, if it is set to zero then free gasdoes not dissolve in undersaturated oil, howeverthis keyword has no effect in models with vapor-ized oil and dissolved gas(VAPOIL and DISGASin RUNSPEC section) which is a case for NorneField, here DRSDT is automatically reset to alarge number.
VAPPARS SCHEDULE used in the systems with VAPOIL and DISGASinstead of DRSDT. User enters two arguments:first represents ability of oil to vaporize and thesecond tendency to get heavier.
Table 4.1: Keywords omitted in Norne Deck
keyword section purpose
WPAVE SCHEDULE controls computation of average pressure in thegrid cells containing well, alternatively it can bedone for particular well block using WWPAVE
GRUPNET SCHEDULE defines well production network structure re-quired when using Network Option
NETBALAN SCHEDULE tunes on Network Option - balancing flow ratesand pressure looses to calculate THP limits for agroup of wells
GECON SCHEDULE allows specification of economic limit data (gas(oil) minimum production rate limits, maximumgas-oil ratio)
Tab 4.2: Kywords from SCHEDULE section omitted in Jewel Suite
4.3. ENKF PLUG-IN TO JEWEL SUITE 23
4.3 EnKF plug-in to Jewel Suite
For the purpose of history matching there is available Ensemble Kalman Filter plug-in in Jewel
Suite created and currently being developed by TNO. There is four subpanels in the main TNO
EnKF History Matching panel. In the Generate ensemble panel it is possible to create initial
ensemble first choosing previously specified realizations of the reservoir properties and the size
of an ensemble with the truth case included if needed. Continue to the next window Setup
EnKF Case several options for the EnKF can be chosen and finally the Control panel allows
to validate the correctness of the ensemble and observations table and to start the simulation.
In the Setup panel user can create or choose an existing EnKF case and link it to the ensem-
ble of interest. Thereafter the parameters to update are chosen from available list. After the
measurement table view is opened the assimilation and observation values for particular wells
can be filled or alternatively a complete table can be simple copied from popular formats as
Excel files. In Edit general settings an important aspects of the simulation can be controlled.
After specification of the choice of the reservoir simulator it need to be declared if the truth
is synthetic and if the reservoir simulation need to be restarted from the beginning after every
time step. Note that it can improve EnKF accuracy however it forbids updating of dynamic
reservoir parameters. Before declaration of the number of ensemble members and simulation
time is specified, in addition, EnkF option need to be selected.
There are four options to choose in the EnKF option field: none, standard, localization or
inflation. The difference between one and standard option is that when the first none is active
it means there is no updates of reservoir parameters in the simulation. This option can be
used to investigate performance of the initial ensemble members without analysis scheme.
The option standard turns on the updates and the theory behind the two remaining choices,
localization and inflation as a methods of reducing spurious correlations was described in
Chapter 2.
4.4 Discrepancies between coarsened and flux based model
Since the observations are available only for E-segment and the size of the grid is an issue
it is convenient to history match only E-segment. Two methods for reducing the simulation
grid, namely coarsening and flux boundary option, were presented previously in this para-
graph. Applying any of these methods brings the question how the performance of the model
changes. This was investigated for Norne Field and the results are presented in the Picture 4.6.
It is shown in the Figure 4.6 how the oil production over the whole field behaves when the
coarse (red line) and the flux (blue dashed line) option is used. The second choice appears
to have the advantage over the coarsened grid since the field production stays closer to the
24 CHAPTER 4. ECLIPSE MODEL IMPORT INTO JEWEL SUITE
Figure 4.6: Difference in FOPR for the coarsened and flux-based simulation.
original results, when the grid size is not reduced (green line). The another benefit is that the
grid in the flux-based simulation consist only of E-segment, therefore its size is smaller than
the coarsened grid, however it requires one additional simulation in order to create the file
with the boundary information. In history matching application it is aimless to perform full
simulation after every update step for every ensemble member to create flux file. In results,
the use of one flux file (for instance averaged over all ensemble members) created at the ini-
tial conditions brings the question about uncertainty on the boundaries for particular members.
The Figure 4.6 suggests the choice of flux option instead of coarse, however there is a major
drawback for this method. The reservoir model can be divided into flux regions in order to
distinguish local properties of the field, for instance the transmissibility multipliers between
flux regions can be specified. When any of the wells is completed in more then one flux region,
Eclipse returns an error message and the simulation is broken. Therefore to compare coarse
and flux options 20 flux regions in Norne was replaced by 2 regions, as seen in the Figures 4.7
and 4.8.
Figure 4.7: Norne with 20 flux regions Figure 4.8: Nornw with 2 flux regions
This change, however, significantly change the field oil production response which can be
visible comparing coarsen option in the Figure 4.6 where 2 region were used and the coarse
model with 20 regions in the Figure 4.10 in the next section. This justifies the choice of the
Norne Coarse model for the purpose of history matching in this thesis.
4.5. DISCREPANCIES AFTER NORNE COARSE MODEL IMPORT TO JEWEL SUITE 25
4.5 Discrepancies after Norne Coarse model import to Jewel Suite
Model imported to Jewel Suite differs from the original build in ECLIPSE due to the changes
made in order to adjust it to Jewel Suite standards. The status of bottom hole pressure for well
E-2H and oil production rate for the field are presented in Figure 4.9 and 4.10, respectively.
There are three major factors expected to be responsible for the discrepancies.
First and also the most influential issue is set of edits made in original deck. Omission of
keywords, especially commands controlling the well net properties (i.e. economic limits) influ-
ence the performance of the wells. For instance, after the production limit is reached the well
is disabled. In addition, some keywords controls bottom hole pressure as an important factor
used to maintenance the reservoir. The trouble keywords are explained in details in the Tables
4.2 and 4.3.
Using Darcy equation ECLIPSE calculates transmissibilities between neighbour cells based
on cell shape permeability and net-to-gross. Although Jewel Suite computes transmiisibilities
also from Darcy equation, both methods differs. The detailed methodology can be found in
software manuals [24] [25]. The default calculation of transmissibility in ECLIPSE is overwrit-
ten by calculations from Jewel Suite. In case when there is expected flow between non adjacent
cells, for instance across the fault, then the non-neighbour connections and the corresponding
transmissibilities are generated as well.
Figure 4.9: Difference in BHP in well E-2H for Norne models
In addition there is another change applied to the original Norne model. As it is explained
in Appendix A, there are two possible grid treatments in ECLIPSE. The original Norne Model
grid was created according to Corner Point Geometry (CP) which allows to build more irregular
shapes of the cells than rectangles. However, Block Countered Geometry (BC) approach has
advantage over CP since it requires less amount of data storage, therefore considering relatively
large Norne Model CP has been switched in Jewel Suite output deck to BC approach. All
the properties are assigned to the single cells and ECLIPSE does not use grid cell dimensions
26 CHAPTER 4. ECLIPSE MODEL IMPORT INTO JEWEL SUITE
Figure 4.10: Difference in FOPR for Norne models
to calculate fluid flow, instead however, the pore volume and transmissibilities are calculated
from the cell shape specifications. Transmissibility between cells depends in addition on their
permeabilities whereas pore volume is calculated based on cell dimensions, porosity and net-
to-gross ratio, PORV = DX ∗DY ∗DZ ∗ PORO ∗NTG.
Chapter 5
EnKF simulation results
5.1 Study workflow
The complete work can be divided into two parts. In the first part the Norne model is im-
ported to Jewel Suite. Because of the necessary manual modifications applied in the imported
ECLIPSE deck the aim was to manually edit the output deck in order to match the model
as close as possible with the original. The results of this procedure can be seen in Chapter
4. In the second part the focus is on the history matching of the model using EnKF plug-in
available in Jewel Suite. The results of particular steps performed in order to achieve this goal
are presented in the following chapter. Firstly the sensitivity of the production on reservoir
parameters change is investigated then the initial ensemble is created and finally the EnKF
simulation is performed.
5.2 Sensitivity analysis
Before the preparation of initial ensemble the decision of chosen parameters to update in EnKF
procedure has to be made. The goal is to find reservoir parameters which are the most sensitive
to the simulation results then create initial ensemble for selected properties. Sensitivity analysis
was carried out for parameters listed below.
• porosity
• permeability
• net-to-gross
• transmissibility multipliers between regions
A base run received as an input initial properties defined in the Norne Model. Thereafter, a
number of simulations for every change of the parameter had been carried out and compared
with the base run. Values of the considered properties have been modified by shifting the
distribution over a field by adding and subtracting constant value for every grid block. The
27
28 CHAPTER 5. ENKF SIMULATION RESULTS
constant value 0.3 was added to transmissibility multipliers. The pattern that has been used
is presented in the Table 5.1.
Property Symbol Mean\Value Edit 1 Edit 2
Porosity poro 0.219 0.5*poro 1.5*poro
PermeabilityX kx 348.74 0.5*kx 2*kx
Net-to-gross ntg 0.773 ntg-0.1 ntg+0.1
transmissibility multiplier tran i 0.01 tran i + 0.2
Table 5.1: Sensitivity investigation pattern.
Porosity is usually the parameter that has a great impact on the performance of the reservoir
production. This relation applies to Norne case as well.
Figure 5.1: Sensitivity of oil production onporosity for well E-2H.
Figure 5.2: Sensitivity of oil production onpermeability for well E-2H.
Among the investigated properties porosity and permeability show the most significant impact
on the reservoir production in E-segment. The results of sensitivity analysis are presented in
the Figures 5.1, 5.2, 5.3 and 5.4. Varying the transmissibility multipliers between the 5 regions
in E-segment and the regions in remaining part of the field did not bring meaningful changes
in the oil production. After one of the multipliers has changed, the very little change in oil
production (see Figure 5.4) can be noticed in one of the wells in E-segment. It can be observed
Figure 5.3: Sensitivity of oil production onnet-to-gross for well E-2H.
Figure 5.4: Sensitivity of oil production ontransmissibility multiplier for well E-2H.
that uncertainty of the oil production rises around 1500 day. This can be caused by water
breakthrough which occurs at that time. The water previously isolated from the production
5.3. HISTORY MATCHING RESULTS 29
(for instance, water injected into wells) gains access to the considered production well. The
water production rate for the well E-2H can be seen in Figure B.5 in Appendix B.
5.3 History matching results
The initial ensemble was created via Sequential Gaussian Simulation which theory is explained
in Chapter 2. According to sensitivity analysis the porosity and the permeability were chosen as
parameters to update. There is 60 ensemble members involved in the simulation. During 2520
time steps (days) parameters are updated 4 times: at 900, 1620, 2160 and 2520 day. After
every update simulation is restarted from time 0 with new parameters as an input. Restart
option forbids updates of dynamic properties, however, it prevents possibility of appearance
of non-physical values and thus it makes simulation more stable. The results in Figure 5.5
Figure 5.5: Initial behaviour of WOPR [sm3/day] on the left and after last update step on the right.
include the oil production per day in 3 production wells located in E-segment of Norne Field.
Initial behaviour of the production can be seen on the left where the blue line represents the
mean of initial ensemble and the red dots correspond to observations used in the assimilation.
It is noticeable that the well E-2H is the most sensitive to the initial ensemble which form the
best spread of the production among all 3 wells. Despite the ensembles with poor spread for
30 CHAPTER 5. ENKF SIMULATION RESULTS
wells E-3H and E-3AH do not embrace the observations, the predicted production appears to
approach the measurements. It is visible especially at the time with a major difference between
ensemble and observations. The results of matching the gas and the water production rates
can be found in Appendix B.
The log-permeability in X direction at the initial time and after every update is presented
in Figure 5.6. The model consists of 22 layers and the results are presented for layer 1. The
values represent the mean over the ensemble members. The updated permeability becomes
Figure 5.6: Mean of LogPermX, initial and after updates: 900, 1620, 2160 and 2520.Location: E-segment, layer 1.
very convenient for the field development since the oil, gas and the production wells are located
at a high permeability region. The next property assumed to be uncertain was porosity. The
effect of the updates is demonstrated in Figure 5.7. The reservoir was assumed to be strongly
Figure 5.7: Mean of porosity, initial and after updates: 900, 1620, 2160 and 2520.Location: E-segment, layer 1.
homogeneous at the first layer what is reflected in the mean of the properties in initial ensem-
ble. However, already third update proofs more local variations. The values of the properties
are rising since the model is being matched to the observations and most of them appear to
be greater then the model prediction.
Chapter 6
Remarks and future improvements
The complete history matching workflow requires to focus on a variety of issues involved, i.e.
when there is a need to adjust the model to different software conditions or the specification of
the initial conditions is obligatory or the observations need to be prepared for the assimilation.
Therefore, it is difficult to avoid contribution of the different science disciplines in application
of mathematical procedure. Not only theoretical expertise but also programming skills and
environmental knowledge (depended on application field) are often inevitable. There is several
matters in Norne history matching workflow worth to investigate in order to improve final
results.
One of the common issue in the Ensemble Kalman Filter is the creation of the initial ensemble.
Usually the knowledge of geology is used to generate multiple stochastic reservoir realizations
representing uncertainty in the model. Popular approach is to use sequential Gaussian simu-
lation, which theory is explained in Chapter 2, to estimate parameters conditioned on the well
data. How the initial ensemble, created this way, influences the EnKF results was investigated
by Lorentzen [26]. For the Norne Field ensemble was create via sequential Gaussian simu-
lation with well logs as an input, however, there is strong relation between Norne geological
properties and the particular layer. Therefore, it is reasonable to involve this information in
the creation of initial ensemble. Despite the model is history matched to the observations it
does not mean that there is no error attached to the set of measurements. Therefore, this
uncertainty need to be assessed.
Since often the measurements are acquired at the well locations, the data is not only sparse
but in addition it is often available for a specific region of the field. To improve the experiment
only such part of the field should be history matched since the updates in the wells have less
impact on fluid flow in the long distance regions. It would be convenient to simply separate
the area of interest, however, the question about the boundary conditions appears. The infor-
mation about the fluid flow through the boundary of the isolated region should be included.
31
32 CHAPTER 6. REMARKS AND FUTURE IMPROVEMENTS
The ECLIPSE simulator allows to use FLUX or COARSEN approach described in Chapter 4
to deal with this issue, however, it does not completely neutralize uncertainty of the inflow of
the fluid to the separated area and its outflow outside the boundary. Hence, it is important to
asses uncertainty related to the fluid exchange.
The seismic data is dense comparing to the well observations and helps to investigate the
layer structure in the reservoir. Therefore, recently the adjusting the EnKF method for seismic
inclusion gain popularity. Several papers present Enemble Kalman Filter adopted for use of
the seismic data. Myrseth [27] presents Hierarchical Ensemble Kalman Filter (HEnKF) and
Dong [28] propose the assimilation of seismic impedance in EnKF and test it on synthetic
case. Moreover, if the seismic is available after history matching period it can be used as a
validation source. For the Norne Field there are 3 available surveys for the years 2001, 2003,
2004 and 2006.
The size and complexity of the real reservoir model bring constraint on a number of sim-
ulations during the research. It is common to use from 40 to 100 ensemble members in the
applications nowadays, however for the Norne reservoir the process becomes still time consum-
ing, even with either FLUX or COARSE reduced grid. On the other hand, flow simulation for
each of the ensemble can be performed simultaneously thus the method is ideal for parallel
computing. According to time and memory constraints, several decisions have to be made.
Except choosing a number of ensemble members and simulation length, also density of the
set of observation and assimilation times need to be specified.
Appendices
33
Appendix A
Short Eclipse and Jewel Suite Guide
A.1 ECLIPSE from Schlumberger
There are five compulsory sections in *DATA file which is an input to ECLIPSE reservoir simu-
lator: RUNSPEC, GRID, PROPS, SOLUTION and SCHEDULE together with three optional:
EDIT, REGIONS and SUMMARY. Each section has particular meaning. It is important to
acknowledge here that following chapter describes construction of ECLIPSE data with respect
to Norne Field Model thus not all of possible keywords in ECLIPSE are mentioned. In fact
there is much more commands that can be used to describe reality more accurate.
Figure A.1: Geometry types: Block Centred and Corner Point,ECLIPSE 100 user course, Schlumberger GeoQuest
The RUNSPEC section contains main characteristics of the model, i.e. unit system, dimen-
sions of the reservoir and input tables dimensions necessary for ECLIPSE to allocate appropriate
amount of memory. The most expensive in memory is a storage of grid information since the
values is attached to every cell. Geometry properties, porosity, permeability and net-to-gross
are converted by ECLIPSE after reading GRID section into a more convenient form for flow
computations. It results in the calculation of pore volume, transmissibility in three directions
and cell center depth for each single cell. Keyword DIMENS is use to specify reservoir dimen-
sions, START is followed by start date of simulation. There are three units systems available
in ECLIPSE: FIELD, METRIC and LAB but there can be only one defined for all data in
the model. Five reservoir phases can be defined in RUNSPEC section: OIL, GAS, WATER,
35
36 APPENDIX A. SHORT ECLIPSE AND JEWEL SUITE GUIDE
DISGAS and VAPOIL.
The last two keywords states for dissolved gas and vapour oil, respectively. For history match-
ing purpose it is worth to mention keywords UNIFIN and UNIFOUT, since the input, as well
as the output files can be either unified or multiple these keywords indicates they are unified
whereas the default option is multiple. When the Ensemble Kalman Filter is applied, after
each assimilation step there is a need to have a new restart files thus this keywords supposed
to be omitted.
The GRID section specifies static properties of the reservoir. In following section grid geom-
etry, porosity, permeability, net-to-gross and aquifer are specified. Based on this informations
ECLIPSE computes midpoint grid depths, pore volumes and block transmissibilities. There
are two types of geometrical description in ECLIPSE. In Block Centred Geometry(BC) blocks
are all rectangular and scheme requires a top depth and cell size in X, Y and Z direction.
Corner Point Geometry (CP) is more complicated and comparing to BC requires much more
data to describe each cell in the grid. A coordinate lines define columns of cells. It can be non
vertical but it is always straight. Two points, one above and the second one below defines
each line. The cells in the column are defined by elevation points as seen in Figure A.1. In
results we need 4 lines and 8 points do describe each cell. This leads to larger data storage
however this geometry allows to describe reality more accurately since grid blocks do not have
to be rectangular any more. All data given for particular cells are read or write in the Universal
Transverse Mercator(UTM) convention for Cartesian grid shown in the Figure A.2. The start
cell read or written is numbered (1, 1, 1) and it is situated at the top, back, left of any display.
The data is read or written in order of X cycling faster followed by Y and Z.
There are two types of cells in the model active and inactive. The second one are excluded
from simulation and flow is not computed for those cells. Although, inputs as porosity, perme-
ability etc. still have to be defined for inactive cells since ECLIPSE need to compute their pore
volumes, depths and transmissibilities. To specify whether cell is active or inactive keyword
ACTNUM is used followed by number 1 or 0 for each active or inactive cell, respectively. All
cell parameters such as PORO (porosity), PERMX, PERMY, PERMZ (permeabilities in x, y
and z directions respectively), NTG(net-to-gross ratio) are averages defined for the center of
the grid block.
The Faults are defined using FAULTS keyword followed by name of the fault, upper and lower
margins of the face and face name (X, Y or Z). The given values are positions of the grid
blocks connected to the fault (see Figure A.3). Keywords MULTFLT can be used to set trans-
MULTIPLY can be used to change these values for defined by user block of cells. Usually
transmissibility multiplier value is between 0 and 1 when the small value refers to small trans-
A.1. ECLIPSE FROM SCHLUMBERGER 37
Figure A.2: Cycling scheme in read and written files,ECLIPSE 100 user course, Schlumberger GeoQuest
missibility. Except transmissibilities for grid blocks connected to faults there is a need to define
this multipliers for the rest of the cells in particular zones according to horizontal barriers. This
is done by use of MULTX, MULTY, MULTZ keywords which states for transmissibility in X,
Y and Z direction, respectively.
The purpose of the GRID section is to calculate pore volumes, cell center depths, transmissi-
bilities and non-neighbour connections. However after ECLIPSE have done its work user can
modify these properties in EDIT section mainly by use of EQUAL and MULTIPLY keywords.
Most of the keywords from GRID section apply to EDIT section.
Figure A.3: Example of faults and transmissibilities definition,ECLIPSE 100 user course, Schlumberger GeoQuest
In the PROPS section tables for PVT data, rock properties, relative permeabilities and cap-
illary pressures are defined. Keywords PVTG, PVTO and PVTW state for gas, oil and water
PVT properties respectively required for material balance calculations at each time step. In
case of PVTG first phase pressure is specified and then vaporized oil-gas ratio, formation vol-
ume factor and gas viscosity at according phase pressure is defined. The PVTO is followed
by description of dissolved gas-oil ratio, the bubble point pressure, oil formation factor and
oil viscosity at the specified bubble point. Finally properties for water are assign, reference
pressure, formation volume factor, water compressibility and viscosity. In following section
rock properties are included simply by using keyword ROCK i.e. compressibility at reference
38 APPENDIX A. SHORT ECLIPSE AND JEWEL SUITE GUIDE
pressure. For the purpose of flow calculation also saturation functions have to be described
i.e. relative permeability and capillary pressure curves in the form of tables.
Figure A.4: Restart methodology,ECLIPSE 100 user course, Schlumberger GeoQuest
In reality parameters in reservoir vary due to the regional geology. In the REGIONS section
reservoir can be subdivide in such areas in order to edit rock and fluid properties i.e. fluids
in place, relative permeability, capillary pressure or fluid contacts. If the focus is only on the
small part of the reservoir, it can be used for reporting purposes as well. Each cell receives a
number of region it is assign to. This number can be between 0 and the maximum number of
regions specified in RUNSPEC section. Commonly used keywords are EQLNUM, PVTNUM,
FLUXNUM and SATNUM to associate different equilibration, PVT properties and saturations
functions for different regions. Fluid-in-place regions are defined by the FIPNUM keyword.
Before simulation begins there is a need to specify initial conditions in the reservoir. This
is done in the SOLUTION section. Initial pressure, phase saturation, oil and gas rates and
analytical aquifer conditions are defined in one of three possible ways: equilibration, enumer-
ation or restart. In the first case ECLIPSE describes gas-oil, oil-water contacts and pressures
automatically base on information in saturation functions from the previous PROPS section.
However, having the sufficient data it is possible to assign initial conditions explicitly for every
cell. This is called enumeration. For the purpose of history matching initial conditions can be
defined in useful RESTART way. After each so called base simulation results are written to
restart file (values are assigned to every cell) and then read as a initial solution for so called
restart run. It is illustrated in the Figure A.4.
Specification of the data output necessary to further plotting procedures is located in SUM-
MARY section. Name construction is explained in the Table A.1. For instance FOPR states for
Field Oil Production Rate or WGIT is Well Gas Injection Total, however, there are exceptions
to this rule, i.e. Well Bottom Hole Pressure is behind WBHP or FWCT is simply Field Water
Cut.
A.1. ECLIPSE FROM SCHLUMBERGER 39
First character Second character Third character Fourth characterF Field O Oil P Production R RateG Group W Water I Injection T TotalR Region G Gas F FlowW Well V VolumeB Block T TracerC Connection L Liquid
Table A.1: Construction of abbreviations in ECLIPSE.
Finally the SCHEDULE section contains well data, i.e. well completion, rate data, well control
and flow correlations for specified time steps. Basically this is the section where the production
and injection of the wells can be managed. The fact that wells are drilled, shut, closed and
opened in different times justifies the purpose of this section. It is very important to honour
the right order of the keywords. Generally this must be consistent with the reality i.e. before
the start of the production of particular well it has to be completed first. Each action at the
certain time step has to be implemented between the specification of the date using DATES
keyword followed by the day description in DAY MONTH YEAR format, for instance 14 AUG
2008 or equally 14 ’AUG’ 2008, and before the next date. There is about sixty keywords
available in this section, albeit only the commonly used will be described in this paragraph.
The keyword WELLSPECS is used to specify the location of the well in the grid and among
other properties, BHP reference depth, flowing phase or fluid PVT table can be described.
It is necessary for each well and has to precede any other action taken on this well and can
be located at the beginning of the schedule section or at any time of the simulation. Using
COMPDAT command all connections between the well and grid block can be defined. Their
status is set to either OPEN or SHUT. However, two keywords WCONPROD and WCON-
INJE can be used to change well status to SHUT or OPEN and to set properties such control
mode, fluid rate limits and bottom hole pressure limits. An alternative is to use WCONHIST
for production wells and WCONINJH for injectors which declares the well as special history
matching well. Defined measured fluid rates can be written to summary files to compare with
simulated values. In case of Norne Filed control mode for producers is set to RESV for most of
the wells what means that ECLIPSE calculate fluid pore volume rates using reservoir pressure
and set the well to produce at that target.
The way how ECLIPSE deals with the flow boundary conditions deserve separate paragraph
attention. If there is a need to isolate part of the field because it is a region of interest it is
possible to reduce simulation time by using USEFLUX keyword. First, the maximum number
of flux regions need to be specified in REGDIMS command in RUNSPEC section, then the
regions are defined in keyword FLUXNUM in the GRID section, where the number of region
to which the cell belongs is assigned to every gridblock. The FLUXNUM regions can be
simply COPY from SATNUM or FIPNUM property. Now the full grid simulation need to be
40 APPENDIX A. SHORT ECLIPSE AND JEWEL SUITE GUIDE
performed once with keyword DUMPFLUX in the grid section what indicates that ECLIPSE
saves the file with the record of the flow moving through the boundary. After the full field
run is finished the DUMPFLUX need to be replaced with the USEFLUX keyword followed by
the name of the flux file. Every time the simulation with USEFLUX is performed it refers
to reduced run. However, in the reduced run the regions of interest need to be set active.
It is done by use of FLUXREG in the grid section followed by the number of active region.
Keywords DUMPFLUX and USEFLUX cannot be used in the same run. The wells outside the
active region are simply omitted in the reduced run, however their rates are read from flux file
and added to the group and field totals. There are two treatments of boundary conditions in
ECLIPSE. As a default flow for each phase is stored in the flux file. However, as an alternative
pressures, saturations, solution gas-oil ratio and vapour oil-gas ratio in the so called halo cells
surrounding the boundary can be saved to flux file. Base on these information in the reduced
run ECLIPSE calculates flows through the flux boundary. To switch between these options
keyword FLUXTYPE in the GRID section can be used followed by either FLUX or PRESSURE.
There is an alternate approach to separate any area from the full field. It involves more
manual edits but does not require creation of the additional files. In the GRID section user
must enter COARSE keyword and specifies not overlapping ”boxes” of cells to consolidate and
the number of output cells. Reserved memory is specified in RUNSPEC section in the LGR
keyword where the number of new cells are entered. After the method is applied, number of
active cells in the model is reduced. Upscaled properties values are assigned to the center cell
in the box and all other cells are set inactive. However, size of this cell is set to the size of the
box thus appropriate transmissibility multipliers can be calculated. All the wells are moved to
the center cell and multilayer connections are reduced to only one. Non neighbour connections
between coarsened grid blocks and between original cells and coarsened are calculated.
A.2 Jewel Suite
JOA Oil & Gas Jewel Suite is a full workflow-integration framework which allows to build and
design from the seismic interpretation, the grid representation to static and dynamic program-
ming and the wells structure. It has a user friendly interface guiding step by step through
preparation of the model and its workflow. Reservoir simulators such SENSOR, ECLIPSE and
IMEX can be pluged in and the results of the simulation can be imported and viewed in the
Jewel Suite.
It is obvious that each cell in the three dimensional grid can be represented by eight cor-
ner points. By contrast, Jewel Suite allows cell to have any number of corner points. Large
reduction of storage data is gained because for the cells having the same vertical indexes only
one cell is created. The K value is then stored in the cell and single depth value at the edges.
Data necessary to build a grid can be either created or imported . Data Creation tools allows
A.2. JEWEL SUITE 41
to create or edit already imported data. In the process of creation tables can be entered
manually or easily copied from formats as Excel files *.xls. If the model already exists but is
created in different application it can be imported. Except previously exported files from Jewel
Suite other formats including ASCII files, grid files, seismic SegY files and well data files are
available. Because of a choice of the simulator in this thesis the focus is on the cooperation
between ECLIPSE and Jewel Suite which can be seen in the Figure NUMBER. It is allowed
to read different type of the data stored in separate files, i.e. *.inc, *.ecl or *.GRDECL. It is
however more convenient to import the ECLIPSE deck described in the previous subsection
containing full set of information required for creation of the model and a simulation.
While the model is created or imported there is a variety of tools in Jewel Suite for even-
tual modifications and analysis. Property Modelling allows to edit existing property of the
field or create a new one. Property Calculator contains a number of functions required such
basic mathematical functions, distribution, interpolation, filtering or upscaling functions. Using
Property Analysis tool user can easily create statistical plots such variograms and histograms
of the particular properties in order to validate quality of data. Wells can be placed in Well
Planning tool and well logs can be imported from file.
Appendix B
Figures
Figure B.1: Initial behaviour of WGPR [sm3/day] on the left and after last update step on the right.
43
44 APPENDIX B. FIGURES
Figure B.2: Initial behaviour of WWPR [sm3/day] on the left and after last update step on the right.
Figure B.3: Mean of LogPermY, initial and after updates: 900, 1620, 2160 and 2520.Location: E-segment, layer 1.
45
Figure B.4: Mean of LogPermZ, initial and after updates: 900, 1620, 2160 and 2520.Location: E-segment, layer 1.
Figure B.5: Water and oil production rates for the well E-2H
46 APPENDIX B. FIGURES
Figure B.6: Initial behaviour of BHP [Bar] on the left and after last update step on the right.
Appendix C
Basic statistics
This section covers the statistical background information required to understand the theory
included in this thesis, however it can be omitted by readers familiar with basic statistics theory.
The purpose of this section is rather to explain the basics than to formulate strict technical
definitions.
Random Variable
It is very convenient to start with the explanation of the term random variable. It is a function
assigning real values to the outcomes of some random experiment called event, X : Ω → Rwith a property that for all x ∈ R we can assign probability for the event X ≤ x. The Ω
is called sample space or probability space. Usually random variables are denoted as capital
letters i.e. X,Y etc. and their values as an according small letters x, y etc, respectively. Thus
expression X = x means that random variable X takes value x.
Cumulative Distribution and Density Function
The cumulative distribution function (cdf) of a random variableX is a function FX : R → [0, 1]
with relation
FX(x) = P (X ≤ x) (C.1)
which gives the probability that X takes values smaller or equal to x. If the cdf is differentiable
then the derivative
fX(x) =dFX(x)
dx(C.2)
is called the probability density function (pdf). It is required for pdf to satisfy conditions
fX(x) > 0 for all x and∫∞−∞ fX(x)dx = 1. The pdf is used to describe the likelihood of a
given random variable. It specifies the probability that the random variable takes a particular
value. If X is a continuous random variable i.e. probability that X takes x at any point x is
47
48 APPENDIX C. BASIC STATISTICS
zero,
P (X = x) = 0 for x ∈ R (C.3)
and if corresponding density fX exists then probability that X lies between a and b is
P (a ≤ X < b) =
∫ b
afX(x)dx (C.4)
The definitions can be extended for several random variables and the joint distribution function